Sparsification-a technique for speeding up dynamic graph algorithms

The authors provide data structures that maintain a graph as edges are inserted and deleted, and keep track of the following properties: minimum spanning forests, best swap, graph connectivity, and graph 2-edge-connectivity, in time O(n^1/2log(m/n)) per change; 3-edge-connectivity, in time O(n^2/3) per change; 4-edge-connectivity, in time O(nα(n)) per change; k-edge-connectivity, in time O(n log n ) per change; bipartiteness, 2-vertex-connectivity, and 3-vertex-connectivity, in time O(n log(m/n)) per change; and 4-vertex-connectivity, in time O(n log(m/n)+nα( n)) per change. Further results speed up the insertion times to match the bounds of known partially dynamic algorithms. The algorithms are based on a technique that transforms algorithms for sparse graphs into ones that work on any graph, which they call sparsification